An experimental study of coupled rotor-body aeromechanical

FOURTH EUROPEAN ROTORCRAFT AND POWERED LIFT AIRCRAFT FORUM

Paper No. 21

~~ EXPERIMENTAL STUDY OF COUPLED ROTOR-BODY AEROOJECHANICAL INSTABILITY OF HINGELESS ROTORS

It\ HuVER

WILLI~'! G. BOUSc!AN AND DEWEY H. HODGES Aeromechanics Laboratory

U.S. Amy Research and Technology Laboratories (AVRADCOM)

Ames Research Center

Moffett Field, California 94035, U.S.A.

September 13-15, 1978 STRESA - ITALY

Associazione Italiana di Aeronautica ed Astronautica Associazione Industrie Aerospaziali

AN EXPERIMENTAL STUDY OF COUPLED ROTOR-BODY AEROMECHANICAL INSTABILITY OF HINGELESS ROTORS

IN HOVER

William G. Bousman and Dewey H. Hodges Aeromechanics Laboratory

U.S. Army Research and Technology Laboratories (AVRADCOM) Ames Research Center

Moffett Field, California 94035, U.S.A. SUMMARY

A 1.62-m diameter, three-bladed rotor model was tested in hover to examine aeromechanical stability of coupled rotor-body systems. Excellent modal frequency data and good lead-lag regressing mode damping data were obtained over a wide range of rotor speeds. Damping data for the body modes were not satisfactory due to nonlinear damping of the gimbal ball bearings. Simulated vacuum testing was performed with circular cross section blades made of tantalum, which resulted in a Lock number 0.2% of the aerodynamic value. The experimental data were compared with theoreti-cal predictions, and the overall agreement was very good.

1. Introduction

Soft inplane hingeless rotor helicopters are susceptible to aero-mechanical instabilities in hover and on the ground, commonly known as air

and ground resonance. Instabilities of this type occur when the rotor lead-lag regressing mode and a body pitch or roll mode become proximate or

coalesce. The classical condition of ground resonance as analyzed by

Coleman and Feingold (Ref. 1) for articulated rotors is a purely mechanical instability, and as such represents a restricted class of problems where coupling between the rotor and body can be simply represented and the effects of aerodynamics ignored. For the hingeless rotor helicopter, however,

coupling between the rotor and body is quite complex, and the rotor aero-dynamics significantly affect system stability. A number of mathematical models of aeromechanical stability have been developed (Refs. 2-4) and these have been used in the design of helicopters with soft inplane rotors. In addition, Boeing Vertol has used small-scale development models to investi-gate parameters affecting aeromechanical stability (Refs. 5-6) and thereby gained confidence in their analyses. Despite these efforts, a comprehensive understanding of aeromechanical stability has not been achieved.

The complexity of coupled rotor-body systems suggests that a valid mathematical model is required in the development of any hingeless rotor helicopter. Although useful in the design and development of a rotorcraft, such a model may be overly cumbersome for use as a general research tool. In particular, if a broad, fundamental understanding of coupled rotor-body stability is desired, then a simplified analytical model is required that can adequately represent the essential features of the coupled rotor-body problem, but is flexible enough to allow wide parameter variation. This is the rationale used by Ormiston in developing the theoretical model dis-cussed in Ref. 7. The same distinction exists between an experimental model used in the development of a new rotorcraft design, and one designed purely as a research tool (e.g., Ref. 8).

The present experiment has been designed to use a relatively simple experimeptal model in a manner analogous to the theoretical model of Ref. 7. The objectives of the experiment are: (1) to obtain a set of data that can be used to validate simple theoretical models of coupled rotor-body

systems, (2) to explore the character of aeromechanical stability of hinge-less rotor helicopters, and (3) to evaluate test techniques suitable for rotor-body stability testing.

The model used in these experiments simulates a hingeless rotor by using a rigid blade with a spring-restrained root hinge in the manner of Ref. 7. However, the hinge location for the experimental model is offset from the rotor center, and as hinge offset is not included in the theoreti-cal model of Ref. 7, the analysis of Ref. 9 is used in the comparison of theory and experiment. The theoretical model of Ref. 9, although developed for the purpose of examining the aeroelastic stability of bearingless

rotors, is also capable of treating the present experimental model with its high. torsional rigidity.

The paper contains, first, a discussion of the theoretical model, and then describes the design of the experiment, including validation pro-cess requirements, model characteristics, and test procedures. Comparisons are then made between theory and experiment, and conclusions are offered. 2. Analytical Model

In this section, the analytical model used to represent the experi-mental model is described. The theory is actually based on a more general analytical model developed for bearingless rotor helicopters and described in Refs. 9 and 10. This discussion includes only elements of the analytical model actually used in the study of air and ground resonance stability of

the experimental model.

The dynamical system consists of two parts, the body and the rotor. The body is assumed to be a rigid body mounted on spring- and damper-restrained hinges in the pitch and roll directions. Translations of the body, although present in the analysis (Ref. 10), are ignored for corre-lation with the experimental data.

A schematic of the body is shown in Fig. 1. Contributions of the hub and mast are included in the mass and inertia properties of the body. The total mass of the body is mf and the moments of inertia for the mass center of the body are Ix and Iy, respectively, for the X and Y directions. The aircraft reference center, shown in Fig. l, is a distance z above the body mass center and a distance h below the hub center. In studying air resonance and ground resonance, in hover and on the ground, respectively, vertical translation and yaw rotation of the body are insig-nificant. In this paper, body roll ~x and body pitch ~y degrees of

freedom are included. Body spring stiffness restrains the 4x and ~y motion. The rotor blades are attached to the hub and rotate at constant

angular velocity Q. The blade has a built-in pitch angle offset relative to the flexbeam 8 . A schematic of one rotor blade is shown in Fig. 2. The analysis has geen developed to treat different pitch-control

configura-tions; however, in the present case the blade is considered to be unre-strained by any form of control system (Fig. 2). In the analysis the flexure or strap is treated as a single uniform beam segment.

3. Theoretical Analysis

A complete derivation of the

equations of motion used in this

analysis is beyond the scope of this paper; the derivation is given in Ref. 10. The analysis is based on the set of generalized forces due to inertia, gravity, body springs (when the aircraft is in ground contact), quasi-steady aerodynamics, and the flexbeam structure. All

these generalized forces (except those due to flexbeam structural loads) can be written exactly,

within these assumptions for the above

analytical model, and analytically linearized about equilibrium. The flexbeam equilibrium deflections can be calculated through a nonlinear

numerical iteration process, and

the flexbeam structural loads for small perturbation deflections can be determined through numerical per-turbation of the equilibrium solu-tion. These steps lead to a system of linear, constant-coefficient, homogeneous, ordinary differential

equations. 3.1 Generalized Forces Fig. l . SHAFT

' r:

-" " N₁ 2 AIRCRAFT 600Y MASS

I

CENTER

I

Z. Nc REFERENCE CENTER

Rotorcraft body model.

Fig. 2. Schematic of blade geometry. In addition to the rigid-body degrees of freedom ~X' ~ , described above, the six degrees of freedom of the kth blade in the rotating system may be ex~res~ed as: (1) three translations of the blade root corresponding

to the n₁, _{n 2,} n~ axis system, uk, vk, Wk• respectively; and (2) three rotations 'k· sk, and ek, lead, flap, and r.itch angles, respectively, of

the bla~e (beginni~g with 'k about t~e n~ axis, Sk about the axis

cos 'kn~ - sin cknl, 'md 8k about nx axis). Such a sequence is noted

only for the purpose of definition; obviously, the sequence of angles will

not affect the solution of the equations of motion.

Exact expressions for the generalized forces due to inertia, gravity, body springs, the pitch-control systems, and quasi-steady airloads are derived in Ref. 10. The total forces acting along the blade may be resolved

into a force and moment acting at the tip of the kth flexbeam. These are exactly balanced by the structural loads that the kth flexbeam exerts on the kth blade. This is true because all external loads acting along the flexbeam are neglected. All of the loads except the structural loads of the flexbeam can be expressed in closed form. A numerical scheme that

solves the equilibrium deflections is described in the next section, treating the flexbeam structural loads implicitly.

3.2 Flexbeam Equilibrium Deflections

The force vector Fk, representing forces that the kth blade exerts on the kth flexbeam, and the moment vector Mk' representing the

moments that the kth blade exerts on the kth flexbeam, have steady and unsteady components:

(1)

Quantities without subscripts are steady components and the tilde quantities are unsteady components. Similarly,

wk = w + wk (t)

(2) 'k r; + 2k ( t)

sk = s + iik(t) t;k = r; + 2k ( t)

The steady components F and M possess geometric nonlinearity in u, v, w, t;,

S,

8. The unsteady components Fk and Mk may be analytically linearized in the following quantities and their first and second deriva-tives: Uk' Vk' Wk' ~k' Sk' Bk' $X, $y'

Consider only the steady part of the solution. If u, v, w, t;,

S,

8 were known (the deflections of the blade and tip of the flexbeam), then

F and M would be determined. These deflections are not known, however. If the forces and moments at the root of the flexbeam FR, MR were known, the geometrically exact expressions for the curvature-slope-deflection relations could be numerically integrated along the flexbeam to obtain the correct tip deflections. Thus, what we have is a type of two-point boundary value problem. The expressions relating the tip deflections to tip forces and moments are known. The relation between root forces and moments and tip deflections is determined through numerical integration.

A workable, accurate scheme for the solution of this problem is given in the following seven steps: (1) assume a set of flexbeam tip deflections ut, vt' wt, r; , S , 8 , (2) calculate the force and moment

F and M based on the asEumea deflections, (3) transfer the force and moment F and M to the equivalent root force and moment FR and MR using the assumed tip deflections, (4) evaluate the tension force and bending and torsion moments for a generic point along the flexbeam in

terms of the deflections u, v, w, r;,

S,

8, each a function of the distance along the deformed flexbeam, (5) using moment-curvature relations, numeri-cally integrate the geometrinumeri-cally exact expressions relating bending and

torsion curvatures to slopes and deflections (Ref. 10), (6) compare the assumed tip deflections ut' vt, wt, ' t ' st. et' with the tip deflections u, v, w, t;,

a,

8 of step (5), (7) place steps (1) through (6) in an

.'1

•

optimization scheme to minimize

3 = (u- ut)2

+

(v- vt)2

+

(w- wt)2

+

(I; - l;t)2

+

(S - St)2

+

(e - e )

2

t

(3)

Naturally, the minimum 3= 0 is the exact solution to the equilibrium position of the rotor blades. This minimum can be found by use of a non-linear least squares algorithm.

3.3 Perturbation Flexbeam Structural Loads

In addition to the equilibrium solution the perturbation forces and moments that the flexbeam exerts in response to perturbation deflections must be determined. This is done by cyclically perturbing the equilibrium force and moment components in the directions of u, v, w, C, 8, 8 and looking at the new deflections. Symbolically, this is expressed as

(ik Fuk

'\

jlvk wk [ F] :Fwk ( 4) = l;k gl;k sk Msk

i\

Me k

where [F] is a 6 x 6 matrix, identical for all k, the elements of which are

deflection due to jth perturbed load) - ith equilibrium deflection

(5)

and s is the amount the loads are perturbed. The matrix elements are symmetric for infinitesimal s. Numerically, however, s cannot be any smaller than the square root of the relative error of the deflections. This will depend on the error criterion in the optimization algorithm. On the other hand, an s that is too large will produce slight nonsymmetry in [F] due to geometric nonlinearity. The force and moment response of the flexbeam to small perturbations of the deflection may be obtained from Eq. (4) by simply inverting [F]. The resulting matrix [F- 1] is the structural stiffness matrix for the flexbeam. To complete the system equations of motion, this matrix may now be combined with the linearized, unsteady perturbation forces and moments produced by the pitch-control system, inertia, gravity, quasi-steady aerodynamics, and the body springs. 3.4 Linearized Perturbation Equations

When all of the generalized forces associated with both blade and body degrees of freedom are linearized in uk, vk, wk, 2k, Sk, ek, ~X' ~Y' and in their first and second time derivatives, there are many terms with coefficients cos ~k and sin ~k· In hover, all of these may be removed

by the multiblade coordinate transformation (Ref. 11), applying this transformation to the present problem are Only the rotor cyclic modes are coupled to the $x, $y hence, the collective and differential collective modes

in the present analysis. The two cyclic modes for six

each, plus the two body modes, yields fourteen degrees

vc, 'Og, We, Ws, ~C! es, Sc, Ss, Eic, 8s, ¢x, ¢yl where

and F F 2 'h

L::

b k=l ' \ cos ljik 2 b

u = - <r"' ilk sin ljik

s b t ' l u c u s 2 b b

J~i

•

.

F sin ljik uk The details of given in Ref. 10. body motion, and are not included

degrees of freedom

of freedom lie, U_{8 ,}

(6)

(7)

These 14 resulting linear ordinary differential equations with

constant coefficients can be formed into a matrix equation

[A]{xl

+

[B]{xl

+

[C]{xl = o (8) where [A], [B], and [C] contain elements that depend on the system parame-ters and blade equilibrium deflections u, v, w,

s,

6, 8, These matrices contain no small angle assumptions or geometric approximations. Equation

(8) is solved as a conventional eigenvalue problem, 4. Mathematical Model Validation

A primary objective of the experiment was to provide data suitable for use in validating mathematical models of coupled rotor-body systems. To be successful, the validation process must meet three requirements:

(l) the experimental model must closely simulate the theoretical model; (2) the parameter variation must be sufficiently extensive to test the limitations of the theory; and (3) the bases of comparison between theory and experiment must be valid.

The requirement that the experimental model simulate the analytical model is necessary to insure that differences between the theoretical

predictions and experimental measurements reflect limitations of the theory and not the experimental modeling process. As an example, the simplified

theoretical model of Ref. 7 assumes that the rotor blades are hinged at the hub centerline and that their stiffness is due to a hinge-mounted spring. In the design of an experimental model it is difficult to place the hinge centerline at the rotor center. Thus, in any comparison of theory and experiment, there will be an effect due to hinge offset that is in the experimental data but not in the theory. The resulting ambiguity is that there is no way to know whether a difference that occurs is due solely to hinge offset, or whether there is an additional source of error.

There are other difficulties that occur in designing an experimental model to match a simplified mathematical model. Of these, joints and bearings present one of the more troublesome problems. Any loose joint, and many kinds of bearings, introduce friction or viscous damping and slop. Par-ticularly at model scale, nonlinearities or excessive damping can drasti-cally affect system damping and must be avoided. A second problem is

that of extraneous system modes. The simplified theoretical model may assume that the blade is rigid or that body motion is constrained purely to pitch and roll. The experimental model, in fact, will have numerous higher modes, and the model designer's problem is to insure that these modes do not couple with the primary modes of interest. In emphasizing

the necessity of the experiment matching the theory, it is also necessary to point out that the simplified theory must provide an adequate represen-tation of potential helicopter designs, or the entire exercise loses its

value.

The second requirement in the validation process is the need to provide for extensive parameter variation in the experimental design. Every parameter that is of significance in the theoretical model defines a dimension in an n-dimensional space that defines the model region of applicability. To validate the theory over the entire n-dimensional space would be difficult and would undoubtedly be a waste of time. However, it

is desirable to select the more significant parameters and provide a suf-ficient range of variation to determine the limitations of the theory. Clearly, comparing theory and experiment for one or two points is of little

value.

The third requirement is that the bases of comparison must be valid. In investigating the aeromechanical stability of a hingeless rotor heli-copter, the behavior of the least damped mode is of the greatest importance; but there are a number of bases of comparison that may be used in the

validation process, and some may be more useful than others. A hierarchy of these bases can be devised, the lowest basis.of which is the easiest to obtain but provides the least confidence in the theory, while the highest basis represents the most difficult experimental problem but provides the greatest confidence in the theory. For aeromechanical stability tests, an appropriate hierarchy starts with frequency measurements of a single mode, frequencies of all significant modes, stability boundaries, modal damping of a single mode and then all significant modes, and finally, mode shape measurements representing the most difficult problem. In selecting appropriate bases for an experiment, it may be necessary to strike a balance between the level of confidence that is needed and the difficulty of the experimental problem.

5. Experimental Model

Experimental data were obtained with a 1.62-m diameter, three-bladed model rotor as shown in Fig. 3. The electrically powered model is mounted on a gimbal frame that allows pitch and roll freedom, and the various modes of the system are excited by an electromagnetic shaker that is attached to the body through a linkage and pneumatic clamp.

The rotor blades were designed to be very stiff, and most of the blade flexibility is concentrated in root flexures as shown in an exploded view in Fig. 4. The root flexures are designed so that the lead-lag and flapping flexibility are in separate flexures. By folding the structure back on itself, the spanwise locations of the flexures are made coincident. The design of the flexures is such that the torsional stiffness is very

Fig. 3. 1.62-m diameter rotor model.

high; at 1000 rpm the blade nondimensional torsion frequency is greater than 20/rev. By designing the blades to be very stiff, and concentrating the major portion of flexibility in the root flexures, the model design approximates a rotor blade with only one flap and one lead-lag degree of freedom. This design provides a good approximation of the theoretical model that assumes all blade flexibility to be concentrated in a flexible element at the blade root. However, some blade flexibility is outboard of the root flexures (Ref. 12), and to this degree the model design differs from the theory. The root flexures are strain-gaged to provide measure-ments of flap and lead-lag bending momeasure-ments and torsional momeasure-ments for each blade. Blade pitch angle changes are made outboard of the flexures.

The blades and root flexures are bolted to a hub that is mounted on a static mast, as shown in the schematic of Fig. 5. The static mast is bolted to the body, which is supported on either end by ball bearings. These ball bearings provide the body with roll freedom and are supported in a gimbal frame that is mounted to the yoke by a second set of ball bearings that provide pitch freedom. The yoke is the upper part of the stand, which provides a rigid support for the model.

The model is excited about the roll axis by an electromagnetic shaker that is connected to the body with a pneumatic clamp. The shaker may be used to oscillate the body at a specific frequency or to provide an input deflection in roll. By opening the pneuQatic clamp, the body is then unconstrained and measurements are made of the transient decay of the

Fig. 4. Exploded view of blade root flexures. PNEUMATIC CLAMP GIMBAL FRAME ~--PITCH AXIS YOKE STANO NOTE:

FOR SIMPLIFICATION SHOWN AS

TWO·BLAOEO ROTOR.

STATIC MAST

YOKE PITCH BEARING

'STANO

Fig. 5. Schematic drawing of rotor model and test stand.

modes that have been excited. For unstable conditions, a snubber mechanism within the stand may be actuated by the operator, or the snubber will automatically lock out body motions in the event that blade bending moments become excessive.

Body stiffness is provided by cantilever-beam springs, one in roll and two in pitch. Each spring includes a slider that allows the working

0

m4uo

'' v

v

T

0 \LOCAl " 200 IO 1!/, • I :0,) 0 0 0 c)0

Fig. 6. Local damping coefficient of gimbal pitch and roll bearings.

length of the beam spring to be changed to adjust the body stiffness, and hence, frequency. The range of body frequencies is from about 2 to 20Hz. This design provides a good match of the theoretical model, in that the

body motions are constrained to purely pitch and roll motions, at least

at the lower body frequencies. At higher frequencies, however, the motion

of the body will include some translational motions due to flexibility

in the static mast.

Body damping is supplied by damping in the gimbal ball bearings. Unfortunately, this damping is nonlinear with amplitude as shown in Fig. 6, where a local damping coefficient is defined based upon the log decrement

per half cycle. The increase in damping with a decrease in amplitude is typical of Coulomb, or dry friction. If it is hypothesized that the source of the dry friction is the rubbing between the balls and the cage of the ball bearing, then the damping moment will be proportional to

(1) the number of balls in the bearing that are restrained by the cage, (2) the friction force between ball and cage, and (3) the radius at which the friction force acts. A solution for the nonlinear equation describing

a single degree of freedom system (Ref. 13) is shown in Fig. 6, where the friction force per ball is selected to match the experimental results at 8i

=

6° for the pitch bearing. The presence of nonlinear damping in the experimental model is a deficiency that would be expected to cause some difficulty in correlating the results of theory and experiment.

Measurements of body motion are made with two accelerometers mounted just below the hub on the static mast and film potentiometer strips on the gimbal frame. The bending moment signals on the rotor are routed through

a set of slip rings at the base of the rotor drive shaft; these are com-bined with the body measurements, 1/rev and 60/rev signals, lubricating oil lines, and cooling water lines and routed off the model in a manner to minimize their interference with body motions. Additional details relative to the model design and instrumentation are contained in Ref. 14.

6. Simulated Vacuum Tests

The stability of coupled rotor-body motions is strongly affected by the blade aerodynamics. In the validation of a theoretical model, it would be desirable to test a model in a vacuum to observe the effect of inertial and structural properties without aerodynamics. However, vacuum testing is a difficult experimental problem. A partial solution is to simulate a vacuum by testing at a reduced Lock number. If the definition of Lock number is modified to include the effect of the profile drag coefficient

4 Cd

p acR (l + ___£_)

I a (9)

The Lock number may be reduced by substituting a blade of circular cross section where the lift curve slope is zero. If the circular cross section blade is assumed to have a uniform density, the Lock number becomes

12PCd

0

(10)

and it is seen that the Lock number is proportional to the radius and inversely proportional to the density of the rod and blade chord. If the constraint is added that blade flap inertia must not change from the aero-dynamic blade flap inertia, then:

12I

= constant (11)

From these relations it appears that material density and blade chord should be increased, and blade radius reduced. It is important to note, however, that other dynamic properties of the rotor will be affected by

these changes; e.g., dimensionless hinge offset increases proportionally to 1/R, while rotor mass increases as l/R 2 . Large decreases in rotor radius will therefore cause a large hinge offset effect~ and reduce the body frequencies due to the effect of the increased rotor mass. In addi-tion, the flexibility of the circular cross section blade cannot be easily matched to the aerodynamic blade flexibility. In the present case, this latter point is unimportant as the theoretical model assumes that the blade is rigid.

The configuration selected for this experiment is a reduction in blade radius from 81 em to 38 em and a selection of tantalum for the blade material. Tantalum has a density greater than lead, but has strength and stiffness characteristics similar to steel. The Lock number for this configuration is reduced to 0.2% of the aerodynamic blade value. A pic-ture of the model with the tantalum blades installed is shown in Fig. 7. 7. Modal Measurement Considerations

The essential problem of the experiment is to understand the behavior of the modes that affect the aeromechanical stability of the coupled rotor-body system. An introduction to the behavior of these modes and how to measure their characteristics can be obtained by examin-ing the case of a rotor mounted on a rigid hub, where each blade has a flap and a lead-lag degree of freedom. For a three-bladed rotor, then, the

Fig. 7. Rotor model with tantalum stub blades installed.

individual blade modes will combine to form six rotor modes: collective flap, collective lead-lag, two cyclic flap, and two cyclic lead-lag modes. The two collective modes will not couple with body pitch and roll motions in hover and may be safely ignored. The frequencies of the cyclic modes

will appear in the fixed system at sum and difference frequencies between

rotor speed and the individual blade natural frequencies as shown in Fig. 8 (where the individual blade natural frequencies in the rotating system are shown as dashed lines). The higher frequency modes that are

based on the sum are referred to as progressing modes, while the lower frequency modes based on the difference are the regressing modes.

For rotor speeds below the resonance of rotor speed and the lead-lag natural frequency (n = w,), rotors may be characterized as stiff inplane.

Rotor speeds beyond the resonance describe soft inplane configurations.

It is in this latter regime that the coupled rotor-body instabilities,

ground and air resonance, normally occur. The experimental problem, then,

is to select body modes with frequencies in the range of the lead-lag regressing mode and observe the behavior of the coupled rotor-body system.

The frequency and damping of the rotor regressing modes and the body pitch and roll modes provide an accurate description of coupled rotor-body behavior, and are suitable bases of comparison between theory and

experiment. To observe these modes directly, however, is difficult, as

neither the rotor regressing nor the body modes correspond directly to a physical coordinate. The body pitch mode includes pitch motion, to be sure, but also includes roll motion and rotor flapping. The body roll mode behaves similarly, while the flap regressing mode is made up of not only flapping, but both body motions as well. Moreover, neither the flap nor the lead-lag regressing mode can be identified from individual blade

bending moment signals. To observe these modes it is necessary to

The multiblade coordinates for a three-bladed rotor include a col-lective coordinate and two cyclic coordinates for both flap and lead-lag degrees of freedom. For a hovering rotor, the flap and lead-lag collective coordinates make up the flap and lead-lag collective modes, respectively. But the lead-lag regressing mode, for example, includes both cyclic lead-lag coordinates, as does the lead-lag progressing mode. Thus, measurements of either cyclic lead-lag coordinate will show the presence of both modes. To

observe the lead-lag regressing mode directly requires a narrow bandpass filter to separate the regressing and progressing modes.

The frequency and damping of the modes of interest are obtained by exciting the mode, and measuring the frequency and damping from the transient decay. In the case of the lead-lag regressing mode, the model was

35 30 25 20 15 FLAP PROGRESSING {\). + pl LEAO-LAG PROGRESSING 1n •wrl ROTATING FLAP { p ) / / / / / / ROTATING / / LEAO·LAG / {._,) /

--/

----..c---

LEAD-LAG - - 7 - REGRESSING / / ill ,,, 11 200 600 n,,pm ~LAP REGRESSING I!""! ol 600 >000

Fig. 8, Uncoupled rotor frequencies in rotating and fixed systems.

oscillated in roll at the regressing mode frequency until the blade motions were sufficiently large, at which point the excitation was cut off and the pneumatic clamp that connects the model to the shaker was opened so that the mode could decay without restraint due to the shaker. The flap regress-ing (in the simulated vacuum case) and the body roll mode could not be excited at their modal frequencies without causing excessive loads due to lead-lag response. This is probably an inherent limitation of the tran-sient decay method whenever there is a significant difference in damping between nearby modes. However, both flap regressing and body roll modes could be excited by deflecting the model in roll and quickly releasing it. Responses of the two modes were then separated using a narrow bandpass filter. The body pitch mode was excited to the extent it participated in motion about the roll axis, and by gyroscopic coupling.

Transient decay records were analyzed for modal frequency using both an online spectrum analyzer, and by playing the signals recorded on analog tape back through a tracking filter. Modal damping was deter-mined from analog tape records using the log RMS amplitude of the tracking

filter as described in Ref. 15.

8, Comparison of Theory and Experiment

The configurati'ons tested that will be discussed here are: (1) tan-talum stub blades with roll freedom, (2) tantan-talum stub blades with both pitch and roll freedom, and (3) aerodynamic blades with pitch and roll freedom. Other configurations tested and the comparison of theory and experiment may be found in Ref. 14. For each configuration tested, the primary parameter variation used was rotor speed. By varying rotor speed from approximately 250 to 1000 rpm, rotor configurations were simulated from stiff inplane (w,/Q ~ 1.7/rev) to soft inplane (wc/Q ~ 0.7/rev). For

specific configurations, secondary parameter variations were included, such as body stiffness (frequency) and blade pitch angle, but these variations

were not extensive.

The experimental model properties required for use in the theoretical

~

model were obtained from nonrotating measurements. The body spring

stiff-nesses in the simulated vacuum case, for instance, were calculated based on

the measured nonrotating body frequencies, using the uncoupled pitch and roll equations, and assuming the blade motion was locked out due to the

re-straint of the blade droop stops. The bending and torsional stiffness of the single element flexbeam of the theoretical model were determined directly from measurements of blade nonrotating frequencies. No adjustments were made

to any of these properties to improve the correlation between the theory and

,,,

UNSTABLE .1

---

LEAO·LAG REGRESSING IN VACUUM .2 Llb"-l--,\,--f,;,--;/;o-.;\;,"",--.i;o,--,;,-,;i;,-;;;;;;-~ _{100 200 300 400 500 600 700 800 900 1000} !l,rpm

Fig. 9. Tantalum stub blades with roll freedom. w$ = 2.60 Hz (flagged symbols

are w~

=

2~56 Hz; shaded area shows

effect 3f halving and doubling nominal body damping), (a) Modal frequencies. (b) Lead-lag regressing mode damping.

experimental data. The input proper-ties used by the theoretical model are described in Ref. 14. An

ex-ception to the above comments was

the specification of body damping. Estimates of the nonlinear gimbal damping from nonrotating tests

necessarily represent some sort of "average" damping, which may or

may not provide a good approxi-mation of the nonlinear effects.

Rather than using such an average,

the body damping was arbitrarily set to 3% of critical, and the effect of this assumption was ex-amined by calculating results at half and double this value. 8.1 Tantalum Stub Blades with

Roll Freedom

The tantalum stub blades were run with just the body roll degree of freedom by locking out the pitch motion of the gimbal frame. This resulted in the model and stand having a first pitch frequency of approximately 27 Hz, which is well separated from the body roll and rotor regressing mode frequencies. By testing with the tantalum stub blades to simu-late a vacuum and locking out the body pitch degree of freedom, it was possible to examine an ex-tremely simple coupled rotor-body system. The theory is compared with the experimental results in Fig. 9 for the case of a

non-rotating body frequency of 2.60 Hz. The plot of modal frequen-cies shows the rotor regressing and body mode frequencies (and a small portion of the flap progressing

The experimental estimates of lead-lag regressing mode frequency were obtained from the 'c cyclic coordinate, following excitation of the lead-lag regress-ing mode, and are indicated by circles. The body roll and flap regressregress-ing mode

frequency measurements were obtained from the roll axis potentiometer,

follow-ing the deflection and release of the model in roll, and are indicated by dia-monds for the body roll mode and triangles for the flap regressing mode.

Because these modes comprise more than one degree of freedom, the names 11_body

roll11

or 11

flap regressing" are only a convenient means of naming the modes, but do not constitute an exact description.

As rotor speed increases, the lead-lag mode decreases in frequency until resonance with the rotor speed is reached at about 505 rpm. Above this reso-nance, the lead-lag regressing mode increases with increasing rotor speed. The body roll mode is also rotor-speed dependent within the range of rotor

speeds tested and increases quickly enough with rotor speed to prevent any

coalescence with the lead-lag regressing mode. Interestingly enough, the flap regressing mode coalesces with the lead-lag regressing mode at about 670 rpm,

and a very weak mechanical instability occurs. This coalescence is shown

scaled up five times in the inset to the figure. This weak instability is

dis-cussed in Ref. 7 and is limited to rotors in a vacuum or, as in this case, in

a simulated vacuum. The correlation between the theory and experiment for

modal frequency is excellent. Even at the expanded scale of the inset, the

correlation is still quite good.

Figure 9(b) shows the lead-lag regressing mode damping as a function of rotor speed. Although attempts were made to estimate the damping of the body roll and flap regressing modes, the nonlinearity in the gimbal ball

bearings was evident in both modes, and no suitable quantitative estimates

could be made. Despite the nonlinear gimbal damping, however, the lead-lag

regressing mode damping was linear and well-behaved, except for a few

con-figurations in the vicinity of stability boundaries (these points are noted when they occur). The experimental damping measurements show considerably

more scatter than in the measurements for frequency, and clearly show the

weak instability at the coalescence of the flap and lead-lag regressing modes. The theoretical prediction of lead-lag regressing mode damping

shows good agreement with the data, except for an overestimation of the

damping level, particularly at higher rotor speeds. Although the nonlinear character of the gimbal bearing damping cannot be included in the theoreti-cal model, the effect of different levels of body damping may be used to examine the sensitivity of the predicted results to body damping. The shaded areas in this figure show the effect of halving and doubling the nominal body damping of 3%. For the most part, the lead-lag regressing mode is insensitive to the amount of body damping, except when the flap and lead-lag regressing mode frequencies are proximate. This suggests that the experimental measurements of lead-lag regressing mode damping are valid, despite the nonlinear gimbal damping. In addition, it does not appear that the difference in damping level between theory and experiment can be ascribed to the nonlinear gimbal damping. The computed modal fre-quencies are not affected by the level of gimbal damping.

The lead-lag regressing mode damping calculated for an actual vacuum is also shown in this figure, and it can be seen that these predictions are significantly less than the simulated vacuum predictions, a difference due to the profile drag damping of the tantalum stub blades (cd = 1.0). The original purpose in doing simulated vacuum tests was to exagine the effects of inertial and structural terms without the influence of blade aerodynamics. Although the effects of residual aerodynamics on the lead-lag regressing mode damping are significant, the character of the true

0

fl. rpm

Fig. 10. Mode shapes for tantalum stub blades with roll freedom (modal frequency from Fig. 9).

The behavior of the coupled

rotor-body modes is more readily

understandable if the mode shapes are examined, The modal frequency plot of Fig. 9(a) from zero to 250 rpm is expanded in Fig. 10, and the calculated mode shapes for

selected rotor speeds are plotted

on the figure. For each mode a

vector is used for each coordinate to show relative amplitude and phase. The cosine flap coordinate,

S ,

has been selected as a zero pfiase reference. If the lowest

frequency mode at zero rotor speed

is examined, it can be seen that it

consists of sine flapping, Bs, and body roll. Because of the sign convention used here, inphase motion of Bs and ¢x is represented by

vectors of opposite phase, and vice

versa. Thus, the motion of this mode shape is the body and rotor moving together in a fashion analo-gous to first-mode bending of a

beam. As the rotor speed is increased

from zero, this motion persists with the addition of cosine flapping. From the mode shapes of this first

mode, it is clear that this is a

combined body-flapping mode, despite

the name 11

flap regressing11 _{that has}

been given to it. The next higher mode in frequency, at zero rotor speed, shows purely cosine flapping. In this case, the rotor disk is tilting back and forth about the pitch axis, its motion reacted by the infinite stiffness of the body pitch coordinate. The third mode at zero rpm con-sists of sine flapping and body roll as in the first case, but now the motions are out of phase (inphase vectors mean out-of-phase motion due to

the sign convention). This kind of motion, where the rotor disk tilts one way and the body tilts the other, is analogous to the second bending mode of a beam. If rotor speed is increased to 50 rpm, this third mode changes character and becomes purely a flapping mode (the flap progressing mode), while the characteristic of out-of-phase sine flapping and body roll motion now becomes the property of the second mode, a mode previously referred to as the "body roll" mode. I t is now possible to see that the body roll and flap-regressing modes both have significant amounts of body and flapping motion in them and that they are distinguished by the phasing of the motions. The flap regressing mode is where the rotor disk tilts with the body and has the lower frequency. The body roll mode is where

the disk tilts in opposition to the body and has the higher frequency. The fourth mode in Fig. 10 shows sine lead-lag motion at zero rp~.

This mode is the lead-lag oscillation of the blades against the restraint of the locked-out pitch degree of freedom. As rotor speed is increased to 50 rpm, the mode is made up of sine and cosine lead-lag motion. At higher rotor speeds this mode takes on the character of pure flapping oscillations,

discussed, now shows mostly lag oscillations and has become the lead-lag regressing mode.

From a mathematical point of view, the various modes are unique and well defined. For instance, the third mode in Fig. 9(a) starts out with primarily sine and cosine flapping (flap progressing), the mode shape

changes as rotor speed is increased, and the character becomes sine and

cosine lead-lag oscillations (lead-lag regressing). Above 250 rpm the mode takes on the behavior of out-of-phase roll and sine flapping motions

(body roll). Despite the different behavior this mode shows at different rotor speeds, it is a single, unique mode, as would be clearly seen if it had been shown on a root locus plot. But, to understand the behavior

of the system from a physical point of view, it is necessary to consider

the physical modal behavior. Thus, the lead-lag regressing "mode" starts out as the fourth mode, but as rotor speed is increased it becomes, pro-gressively, the third, second, and finally, the first mode. An element of

confusion is probably inherent in the process of referring to separate

mathematical modes as a single physical 11_mode,11 _{but the process is}

essen-tial to understanding the system behavior.

An additional case was run with the tantalum stub blades and roll freedom alone, which included excitation of the lead-lag progressing mode as well as the lead-lag regressing mode. The nonrotating body roll fre-quency was lowered to 1.89 Hz for this case, and the theory and experiment are compared in Fig. 11. The behavior of the rotor regressing and the body roll modes is similar to that shown in Fig. 9. The agreement between the theory and experiment for modal frequency is again excellent, while the agreement is good for lead-lag regressing damping. Lead-lag progressing mode frequency and damping were obtained by oscillating the model in roll at the progressing mode frequency and estimating the frequency and damping

from the transient decay. These data are shown in the figure as solid

diamonds. A fifth mode is added to the collection described previously.

This mode starts out as the lead-lag progressing mode, and in the vicinity

of 400 rpm the mode becomes the flap progressing mode. The fourth mode, which is the flap progressing mode below 400 rpm, turns into the lead-lag progressing mode. The agreement between the predicted lead-lag progressing frequency and the measurements is very good below 400 rpm but is degraded at higher rotor speeds. Similarly, the prediction of modal damping is good below 400 rpm, but at higher rotor speeds the damping is significantly underpredicted.

If the pitch degree of freedom is added to the analysis, and the nonrotating measurement of pitch frequency is used to specify the pitch stiffness, then the agreement between theory and experiment is much improved, as shown by the dashed lines in Fig. 11. The theoretical pre-diction of the lead-lag progressing mode frequency now shows the same trend as the data, although the agreement is not as good as for the low-frequency modes. The agreement between theory and experiment for the modal damping is not as good as in the frequency case, but it seems clear that the pitch degree of freedom is responsible for the rapid rise in damping that is seen for rotor speeds above 400 rpm. Unlike for the lead-lag regressing mode, the lead-lag progressing mode damping is very sensitive to the amount of damping in the roll gimbal bearings, and this may be responsible for some of the difference between theory and experiment in Fig. ll(b). It is noted that the theoretical predictions for the body roll and rotor regressing mode damping and frequency are the same whether the pitch degree of freedom is included or not. This demonstrates that

'""·Hz LEAD·LAG PROGRESSING --._.,.~!¥--"{;(PZ"""..&-<1 FLAP '--::::::--:-!::C""""'--:C:;:--:::. R E GR E SSI NO 11. rpm - 8 -.7 - 6 -.5

WITHOUT PITCH FREEDOM WITH PITCH FREEDOM t PROGRESSING MODE DAMPING 0 REGRESSING MODE DAMPING

I I + I I I I • • I I

+

I

•

UNSTABLE lbl 1 o~-c,:!:oo:;--:-"!::o-cs~o;co -::,!::oo:-:,c!o·oo ll.1pm

Fig. 11. Tantalum stub blades with roll freedom including progressing modes; w~x = 1.89 Hz. (a) Modal frequencies. (b) Lead-lag mode damping.

tests with a single body degree of freedom can be successfully performed if the frequency of the extra body mode is placed sufficiently far away from the frequency of the modes of interest.

8.2 Tantalum Stub Blades with Pitch and Roll Freedom

The model was tested with the tantlum stub blades, and with both pitch and roll freedom. The nonrotating pitch and roll frequencies were selected to have approximately the same value. A comparison of theory and experiment for modal frequency and damping is shown in Fig. 12.

The major change that results with the addition of the body pitch mode is that the lead-lag regressing and body pitch modes coalesce, and a severe mechanical instability of the ground resonance type occurs at about 860 rpm. Interestingly enough, the weak instability and frequency coalescence between the flap and lead-lag regressing modes, which was seen in the roll freedom alone configuration, is now absent. The correlation between theory and experiment for modal frequency is quite good, although it appears from the experimental data that the body pitch frequency approaches

coales~ence with the lead-lag regressing mode frequency at lower rotor speeds than predicted by theory.

The nonlinear damping in the

gimbal bearings causes estimates of

flap regressing or body mode

damp-ing to be unsuitable, as in the roll

freedom alone case. The lead-lag regressing mode damping for the most

part is insensititve to the level of

body damping over the rotor speed range tested, except in the vicinity of 400 rpm, where the flap and lead-lag regressing modes are strongly coupled, and near the stability boundary. In the vicinity of the stability boundary a number of test points show apparent nonlinearities.

These are indicated in the figure

by an arrow that extends from the initial damping level to the final damping level measured at the end of a data record. The nonlinearity

in damping in this case is in the

same sense as the gimbal bearing damping, that is, the damping de-creases as amplitude inde-creases.

In general, the agreement between

theory and experiment for the lead-lag regressing mode damping is good.

The data scatter is reduced from the

roll freedom alone case, but the reduction is exaggerated by the con-densation of the ordinate scale.

The addition of the body pitch mode complicates the behavior of the various modes. The flap

re-gressing mode is essentially as

before: it is made up of the disk tilting in the same direction as the

,,,

LEAD· LAO REGRESSING I\ LEAD·LAG REGRESSING "' n u

'""!.

o~-r--~F+--+-'-+--+--+--+-1+--~' UNSTABLE s I b) '~,--~,oo~,,~oo~J~o~o~•oo~~~~~soo~~,~oo~~,oo~~~~~~~oco H,rpm

Fig. 12. Tantalum stub blades with pitch and roll freedom; w~x = 2.55 Hz,

w~y = 2.58 Hz (shaded area shows effect of halving and doubling nominal body damping). (a) Modal frequencies.

(b) Lead-lag regressing mode damping.

body tilt, and the two travel together around the azimuth at the flap re-gressing mode frequency. The body pitch and roll modes both represent out-of-phase motio'n, with the body tilting in the opposite direction from the rotor disk. The major difference between the two modes is that the pitch mode shows relatively more body deflection than flapping deflection, while

the roll mode shows more flapping deflection than body deflection. Both body modes have considerable pitch and roll motion in them, although the pitch mode has more pitch deflection than roll deflection, and the roll mode has more roll deflection than pitch deflection.

8.3 Aerodynamic Blades with Pitch and Roll Freedom

The model was tested with the aerodynamic blades installed at blade pitch angles of 0' and 8.9'. The same body springs were used for the pitch and roll axes as for the tantalum stub blade case, but because of the reduced rotor mass there was an increase in the nonrotating body frequencies from 2.58 to 2.62 Hz for pitch, and 2.55 to 2.75 Hz for roll. The first blade pitch angle case is representative of a ground resonance condition as there

LEAD. LAG REGRESSING 1;-J ·

is very little thrust on the rotor (there is some thrust due to blade camber). The modal frequencies and lead-lag regressing mode damping for this condition are shown in Fig. 13. The modal frequencies in this case illustrate a number of differences with the comparable tantalum stub blade

FLAP configuration. Because the hinge

off-REGRESSING IS!

,,,

- 6 -.4 ,· .2 UNSTASLE LEAD· LAG REGRESSING

set is significantly reduced, the lead-lag regressing mode frequency varies more rapidly with rotor speed than before. The flap regressing mode is now heavily damped by the blade aero-dynamics, and frequency information could not be obtained with transient excitation techniques. The body pitch and roll mode frequencies are virtually independent of rotor speed, and cross the lead-lag regressing mode rather than coalescing with it. The lead-lag regressing mode is unstable over a fairly wide range in rotor speed, but the instability is considerably milder than in the tantalum blade case. This tendency of aerodynamics to reduce the

8 lbl severity of the instability as compared

o 1oo 2oo Joo <~oo ₁₁~~~m sao 1oo sao goo 1000 to the vacuum case is noted in Ref. 7, Fig. 13. Aerodynamic blades with

pitch and roll freedom; wtx

=

2.73

Hz, w~y

=

2.62 Hz,

eb

=

0 .

(Shadea area shows effect of halv-ing and doublhalv-ing nominal body damping.) (a) Modal frequencies.

(b) Lead-lag regressing mode damping.

The correlation between the theoretical predictions of frequency and the measure-ments is not as good as for the simulated vacuum conditions. The body mode fre-quencies show slight areas of disagree-ment, at both high and low rotor speeds. The lead-lag regressing mode frequency shows a systematic difference that is greatest at the maximum rotor speed and may reflect the effect of the model blade flexibility. In the rotor speed range where the system is unstable, significant differences appear between the calculated and measured body frequencies, but this may be due to the difficulty in making accurate measurements of the frequency of a stable mode in the

presence of an unstable one. Overall, the correlation for the modal

frequen-cies is good.

The theoretical calculation of lead-lag regressing mode damping shows good agreement with the data at rotor speeds up to the lower stability boundary. Beyond this point the theory underestimates the amount of unstable damping, and as rotor speed is increased the theory overpredicts the recovery in

damp-ing. Although the theory shows that the lead-lag damping is sensitive to the level of body damping in the unstable rotor speed range and beyond, this is not enough to explain the observed differences. The theoretical model uses a symmetrical airfoil, and so for these conditions there is no inflow. The experimental model, however, employs a cambered airfoil and in this respect fails to match the theory. The inexact match between theory and experiment results in an ambiguity-it is not known how much of the difference can be explained by the effects of camber and inflow. As in the tantalum stub

blade configuration, there are experimental measurements that show apparent nonlinearities in damping in the unstable regime. The test points at 725 rpm show damping variation with amplitude that has the same sense as the gimbal bearings, that is, as amplitude increases, the damping becomes less. What is more interesting is that at 800 rpm the nonlinear damping is opposite to the sense seen in the gimbal bearings. The reason for these nonlinearities is unknown.

The mode shapes in the aerodynamic case are not significantly dif-ferent from the tantalum stub blade case. Perhaps most interesting is how the various flapping-body modes are affected by the increase in aerodynamic flap damping. The flap regressing mode has become heavily damped, and this seems reasonable as the mode is made up of inphase flapping and body

motions. Thus, tilt of the disk is accentuated by tilt of the body, and the normal flapping velocity is increased by the body rotation. For the body modes the rotor disk motion is opposite in phase to the body motion, and the effect of the flap damping is significantly reduced. In effect the flapping velocity is decreased by the opposite motion of the body, It is these body modes that couple with the lead-lag regressing mode to cause aeromechanical instabilities of the air and ground resonance type. The fact that the body motions are

out of phase with blade flapping may explain why techniques that are effective in augmenting lead-lag damping for rigid hub conditions lose all effectiveness when body freedom is added (Ref. 16).

The theory is compared with the experiment for the high thrust case where eb = 8.9° in Fig. 14. Experimental measurements were made for rotor speeds at 550 rpm and above. The modal frequencies are similar to the low thrust case, and the same systematic difference between theory and experiment is seen for the lead-lag regressing mode. The calculated values for body roll mode frequency are less than the measured values, but in general, the correlation is good.

The lead-lag regressing mode damping is considerably changed from the low thrust case. The effect of blade pitch angle and thrust is to significantly increase the damping at rotor speeds below the stability boundary, and to increase the size and the severity of the unstable region (Ref. 7). The theory shows reasonable agreement with the data, and in particular calculates the amount of unstable damping quite well. However, it underestimates the recovery in the damping at the upper stability boundary, a trend that is opposite to that seen in

- 8

e

0 0 0 LEAD-LAG REGRESSING (i) UNSTABLE 8 lbl 8 !c, ::::.__,.:, oo;;--,,;!;oo,.-3;-;ioo;;---;,;!;oo,.-.Jsoo""'j,",.-,J'oo;;--;8;!,oo;-900;;/;;;--;;;i, ooo n.rpm

Fig. 14. Aerodynamic blades with pitch and roll freedom; W$x = 2.73 Hz,

W$y = 2.62 Hz, eb = 8.9°. (Shaded area shows effect of halving and doubling nominal body damping.) (a) Modal frequen-cies. (b) Lead-lag regressing mode

the low thrust case. Apparent nonlinearities in damping again appear in the unstable region, and as in the low thrust case, the sense of the

non-linearity is the same as the gimbal bearing nonnon-linearity at the lower speed and opposite in sense at the higher speed.

9. Concluding Remarks

The experiment reported in this paper was undertaken to obtain data

for the validation of simplified theoretical models of coupled rotor-body

systems, and to obtain a better understanding of hingeless rotor helicopter

aeromechanical stability. The following conclusions are offered:

(1) Modal frequency data of excellent quality were obtained for the lead-lag regressing mode, the body pitch and roll modes, and in the simu-lated vacuum case for the flap regressing mode. Modal damping data for the

lead-lag regressing mode were also obtained and were of good quality.

(2) Nonlinear damping characteristics of the gimbal ball bearing prevented acquiring adequate modal damping data for the flap regressing

or body modes. No significant effect on the modal frequency measurements

or lead-lag regressing mode damping due to the gimbal damping was observed. (3) Both theory and experiment show a weak mechanical instability

to occur when the flap and lead-lag regressing mode frequencies coalesce

under simulated vacuum conditions and with roll freedom alone. This

insta-bility was predicted in Ref. 7.

(4) Both theory and experiment demonstrate a mechanical instability

of the ground resonance type for the simulated vacuum tests with pitch and

roll freedom.

(5) Both theory and experiment show aeromechanical instabilities for

tests with the aerodynamic blades installed with pitch and roll freedom.

Aeromechanical instability occurs at both low and high thrust conditions.

(6) The correlation between theory and experiment for modal frequency

is in general very good. The correlation for lead-lag regressing mode

damping is generally good. Agreement between experiment and theory is better for the simulated vacuum tests than with the aerodynamic blades installed.

(7) The use of circular cross-section blades of tantalum to simulate vacuum testing is a useful technique. Despite some residual effects due to profile drag, the tantalum stub blades used in the experiment provide a good approximation of the behavior of a coupled rotor-body system in a vacuum.

(8) The progression in test configurations from simulated vacuum with only one body degree of freedom, to an aerodynamic blade configuration with both pitch and roll freedom, provided an opportunity to observe the funda-mental behavior of coupled rotor-body systems as complexity was increased.

(9) Calculation of mode shapes provided basic knowledge of the behavior of the model. The flap regressing and body modes all contain substantial

flapping and body motion. The flap regressing mode is distinguished from the body modes by the inphase tilting of the disk with the body (like the first bending mode of a beam). With the inclusion of aerodynamics this mode is highly stabilized. The body modes are characterized by out-of-phase